Abstract
In the present paper we construct all short representation of so(3, 2) with the sl(2, ℂ) symmetry made manifest due to the use of sl(2, ℂ) spinors. This construction has a natural connection to the spinor-helicity formalism for massless fields in AdS4 suggested earlier. We then study unitarity of the resulting representations, identify them as the lowest-weight modules and as conformal fields in the three-dimensional Minkowski space. Finally, we compare these results with the existing literature and discuss the properties of these representations under contraction of so(3, 2) to the Poincare algebra.
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S. Weinberg, Photons and gravitons in S-matrix theory: derivation of charge conservation and equality of gravitational and inertial mass, Phys. Rev. 135 (1964) B1049 [INSPIRE].
S.R. Coleman and J. Mandula, All possible symmetries of the S matrix, Phys. Rev. 159 (1967) 1251 [INSPIRE].
X. Bekaert, N. Boulanger and P. Sundell, How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples, Rev. Mod. Phys. 84 (2012) 987 [arXiv:1007.0435] [INSPIRE].
E.S. Fradkin and M.A. Vasiliev, On the gravitational interaction of massless higher spin fields, Phys. Lett. B 189 (1987) 89 [INSPIRE].
M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 243 (1990) 378 [INSPIRE].
M.A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dSd, Phys. Lett. B 567 (2003) 139 [hep-th/0304049] [INSPIRE].
S. Giombi and X. Yin, Higher spin gauge theory and holography: the three-point functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].
N. Boulanger, P. Kessel, E.D. Skvortsov and M. Taronna, Higher spin interactions in four-dimensions: Vasiliev versus Fronsdal, J. Phys. A 49 (2016) 095402 [arXiv:1508.04139] [INSPIRE].
M.A. Vasiliev, Current interactions and holography from the 0-form sector of nonlinear higher-spin equations, JHEP 10 (2017) 111 [arXiv:1605.02662] [INSPIRE].
V.E. Didenko, O.A. Gelfond, A.V. Korybut and M.A. Vasiliev, Spin-locality of η2 and \( {\overline{\eta}}^2 \) quartic higher-spin vertices, JHEP 12 (2020) 184 [arXiv:2009.02811] [INSPIRE].
E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [Erratum ibid. 660 (2003) 403] [hep-th/0205131] [INSPIRE].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, Towards holographic higher-spin interactions: four-point functions and higher-spin exchange, JHEP 03 (2015) 170 [arXiv:1412.0016] [INSPIRE].
X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, Quartic AdS interactions in higher-spin gravity from conformal field theory, JHEP 11 (2015) 149 [arXiv:1508.04292] [INSPIRE].
C. Sleight and M. Taronna, Higher spin interactions from conformal field theory: the complete cubic couplings, Phys. Rev. Lett. 116 (2016) 181602 [arXiv:1603.00022] [INSPIRE].
C. Sleight and M. Taronna, Higher-spin gauge theories and bulk locality, Phys. Rev. Lett. 121 (2018) 171604 [arXiv:1704.07859] [INSPIRE].
D. Ponomarev, A note on (non)-locality in holographic higher spin theories, Universe 4 (2018) 2 [arXiv:1710.00403] [INSPIRE].
R.R. Metsaev, Poincaré invariant dynamics of massless higher spins: fourth order analysis on mass shell, Mod. Phys. Lett. A 6 (1991) 359 [INSPIRE].
R.R. Metsaev, S matrix approach to massless higher spins theory. 2: the case of internal symmetry, Mod. Phys. Lett. A 6 (1991) 2411 [INSPIRE].
D. Ponomarev and E.D. Skvortsov, Light-front higher-spin theories in flat space, J. Phys. A 50 (2017) 095401 [arXiv:1609.04655] [INSPIRE].
C. Devchand and V. Ogievetsky, Interacting fields of arbitrary spin and N > 4 supersymmetric selfdual Yang-Mills equations, Nucl. Phys. B 481 (1996) 188 [hep-th/9606027] [INSPIRE].
D. Ponomarev, Chiral higher spin theories and self-duality, JHEP 12 (2017) 141 [arXiv:1710.00270] [INSPIRE].
E.D. Skvortsov, T. Tran and M. Tsulaia, Quantum chiral higher spin gravity, Phys. Rev. Lett. 121 (2018) 031601 [arXiv:1805.00048] [INSPIRE].
E. Skvortsov, T. Tran and M. Tsulaia, More on quantum chiral higher spin gravity, Phys. Rev. D 101 (2020) 106001 [arXiv:2002.08487] [INSPIRE].
E. Skvortsov and T. Tran, One-loop finiteness of chiral higher spin gravity, JHEP 07 (2020) 021 [arXiv:2004.10797] [INSPIRE].
E. Skvortsov, T. Tran and M. Tsulaia, A stringy theory in three dimensions and massive higher spins, Phys. Rev. D 102 (2020) 126010 [arXiv:2006.05809] [INSPIRE].
A.K.H. Bengtsson, A Riccati type PDE for light-front higher helicity vertices, JHEP 09 (2014) 105 [arXiv:1403.7345] [INSPIRE].
E. Conde, E. Joung and K. Mkrtchyan, Spinor-helicity three-point amplitudes from local cubic interactions, JHEP 08 (2016) 040 [arXiv:1605.07402] [INSPIRE].
S. Ananth, Spinor helicity structures in higher spin theories, JHEP 11 (2012) 089 [arXiv:1209.4960] [INSPIRE].
A.K.H. Bengtsson, Notes on cubic and quartic light-front kinematics, arXiv:1604.01974 [INSPIRE].
D. Ponomarev, Off-shell spinor-helicity amplitudes from light-cone deformation procedure, JHEP 12 (2016) 117 [arXiv:1611.00361] [INSPIRE].
N. Boulanger, S. Leclercq and P. Sundell, On the uniqueness of minimal coupling in higher-spin gauge theory, JHEP 08 (2008) 056 [arXiv:0805.2764] [INSPIRE].
E.D. Skvortsov, On (un)broken higher-spin symmetry in vector models, in International workshop on higher spin gauge theories, World Scientific, Singapore (2015) [arXiv:1512.05994] [INSPIRE].
R.R. Metsaev, Light-cone gauge cubic interaction vertices for massless fields in AdS4, Nucl. Phys. B 936 (2018) 320 [arXiv:1807.07542] [INSPIRE].
E. Skvortsov, Light-front bootstrap for Chern-Simons matter theories, JHEP 06 (2019) 058 [arXiv:1811.12333] [INSPIRE].
A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat holography: aspects of the dual field theory, JHEP 12 (2016) 147 [arXiv:1609.06203] [INSPIRE].
S. Pasterski, S.-H. Shao and A. Strominger, Gluon amplitudes as 2d conformal correlators, Phys. Rev. D 96 (2017) 085006 [arXiv:1706.03917] [INSPIRE].
A. Schreiber, A. Volovich and M. Zlotnikov, Tree-level gluon amplitudes on the celestial sphere, Phys. Lett. B 781 (2018) 349 [arXiv:1711.08435] [INSPIRE].
L. Ciambelli, C. Marteau, A.C. Petkou, P.M. Petropoulos and K. Siampos, Flat holography and Carrollian fluids, JHEP 07 (2018) 165 [arXiv:1802.06809] [INSPIRE].
L. Donnay, A. Puhm and A. Strominger, Conformally soft photons and gravitons, JHEP 01 (2019) 184 [arXiv:1810.05219] [INSPIRE].
S. Stieberger and T.R. Taylor, Symmetries of celestial amplitudes, Phys. Lett. B 793 (2019) 141 [arXiv:1812.01080] [INSPIRE].
W. Fan, A. Fotopoulos and T.R. Taylor, Soft limits of Yang-Mills amplitudes and conformal correlators, JHEP 05 (2019) 121 [arXiv:1903.01676] [INSPIRE].
T. Adamo, L. Mason and A. Sharma, Celestial amplitudes and conformal soft theorems, Class. Quant. Grav. 36 (2019) 205018 [arXiv:1905.09224] [INSPIRE].
B. Nagaraj and D. Ponomarev, Spinor-helicity formalism for massless fields in AdS4, Phys. Rev. Lett. 122 (2019) 101602 [arXiv:1811.08438] [INSPIRE].
B. Nagaraj and D. Ponomarev, Spinor-helicity formalism for massless fields in AdS4. Part II. Potentials, JHEP 06 (2020) 068 [arXiv:1912.07494] [INSPIRE].
B. Nagaraj and D. Ponomarev, Spinor-helicity formalism for massless fields in AdS4. Part III. Contact four-point amplitudes, JHEP 08 (2020) 012 [arXiv:2004.07989] [INSPIRE].
K.I. Bolotin and M.A. Vasiliev, Star product and massless free field dynamics in AdS4, Phys. Lett. B 479 (2000) 421 [hep-th/0001031] [INSPIRE].
J.M. Maldacena and G.L. Pimentel, On graviton non-Gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].
T. Adamo and L. Mason, Einstein supergravity amplitudes from twistor-string theory, Class. Quant. Grav. 29 (2012) 145010 [arXiv:1203.1026] [INSPIRE].
N. Colombo and P. Sundell, Higher spin gravity amplitudes from zero-form charges, arXiv:1208.3880 [INSPIRE].
V.E. Didenko and E.D. Skvortsov, Exact higher-spin symmetry in CFT: all correlators in unbroken Vasiliev theory, JHEP 04 (2013) 158 [arXiv:1210.7963] [INSPIRE].
O.A. Gelfond and M.A. Vasiliev, Operator algebra of free conformal currents via twistors, Nucl. Phys. B 876 (2013) 871 [arXiv:1301.3123] [INSPIRE].
D. Skinner, Twistor strings for N = 8 supergravity, JHEP 04 (2020) 047 [arXiv:1301.0868] [INSPIRE].
A. David, N. Fischer and Y. Neiman, Spinor-helicity variables for cosmological horizons in de Sitter space, Phys. Rev. D 100 (2019) 045005 [arXiv:1906.01058] [INSPIRE].
P.A.M. Dirac, A remarkable representation of the 3 + 2 de Sitter group, J. Math. Phys. 4 (1963) 901 [INSPIRE].
E.D. Skvortsov and M.A. Vasiliev, Geometric formulation for partially massless fields, Nucl. Phys. B 756 (2006) 117 [hep-th/0601095] [INSPIRE].
D.S. Ponomarev and M.A. Vasiliev, Frame-like action and unfolded formulation for massive higher-spin fields, Nucl. Phys. B 839 (2010) 466 [arXiv:1001.0062] [INSPIRE].
M.V. Khabarov and Y.M. Zinoviev, Massive higher spin fields in the frame-like multispinor formalism, Nucl. Phys. B 948 (2019) 114773 [arXiv:1906.03438] [INSPIRE].
M.A. Vasiliev, Unfolded representation for relativistic equations in (2 + 1) anti-de Sitter space, Class. Quant. Grav. 11 (1994) 649 [INSPIRE].
A.V. Barabanshchikov, S.F. Prokushkin and M.A. Vasiliev, Free equations for massive matter fields in (2 + 1)-dimensional anti-de Sitter space from deformed oscillator algebra, Theor. Math. Phys. 110 (1997) 295 [Teor. Mat. Fiz. 110N3 (1997) 372] [hep-th/9609034] [INSPIRE].
O.V. Shaynkman and M.A. Vasiliev, Scalar field in any dimension from the higher spin gauge theory perspective, Theor. Math. Phys. 123 (2000) 683 [Teor. Mat. Fiz. 123 (2000) 323] [hep-th/0003123] [INSPIRE].
C. Iazeolla and P. Sundell, A fiber approach to harmonic analysis of unfolded higher-spin field equations, JHEP 10 (2008) 022 [arXiv:0806.1942] [INSPIRE].
N. Boulanger, C. Iazeolla and P. Sundell, Unfolding mixed-symmetry fields in AdS and the BMV conjecture. Part I. General formalism, JHEP 07 (2009) 013 [arXiv:0812.3615] [INSPIRE].
J.A. de Azcarraga, S. Fedoruk, J.M. Izquierdo and J. Lukierski, Two-twistor particle models and free massive higher spin fields, JHEP 04 (2015) 010 [arXiv:1409.7169] [INSPIRE].
N. Boulanger, D. Ponomarev, E. Sezgin and P. Sundell, New unfolded higher spin systems in AdS3 , Class. Quant. Grav. 32 (2015) 155002 [arXiv:1412.8209] [INSPIRE].
Y.M. Zinoviev, Massive higher spins in d = 3 unfolded, J. Phys. A 49 (2016) 095401 [arXiv:1509.00968] [INSPIRE].
E. Angelopoulos, M. Flato, C. Fronsdal and D. Sternheimer, Massless particles, conformal group and de Sitter universe, Phys. Rev. D 23 (1981) 1278 [INSPIRE].
E. Angelopoulos and M. Laoues, Masslessness in n-dimensions, Rev. Math. Phys. 10 (1998) 271 [hep-th/9806100] [INSPIRE].
X. Bekaert, Singletons and their maximal symmetry algebras, in 6th summer school in modern mathematical physics, (2011) [arXiv:1111.4554] [INSPIRE].
N.G. Misuna, Off-shell higher-spin fields in AdS4 and external currents, arXiv:2012.06570 [INSPIRE].
E. Majorana, Relativistic theory of particles with arbitrary intrinsic angular momentum, Nuovo Cim. 9 (1932) 335 [INSPIRE].
P.A.M. Dirac, Unitary representations of the Lorentz group, Proc. Roy. Soc. Lond. A 183 (1945) 284.
I.M. Gelfand and M.A. Naimark, Unitary representations of the Lorentz group, Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947) 411.
Harish-Chandra, Infinite irreducible representations of the Lorentz group, Proc. Roy. Soc. Lond. A 189 (1947) 372.
V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals Math. 48 (1947) 568 [INSPIRE].
I.M. Gelfand and A.M. Yaglom, General relativistically invariant equations and infinite-dimensional representations of the Lorentz group, Zh. Eksp. Teor. Fiz. 18 (1948) 703 [INSPIRE].
I.M. Gel’fand, M.I. Greav and N.Y. Vilenkin, Generalized functions, volume 5, Academic Press, U.S.A. (1966).
S. Fedoruk and V.G. Zima, Covariant quantization of d = 4 Brink-Schwarz superparticle with Lorentz harmonics, Theor. Math. Phys. 102 (1995) 305 [Teor. Mat. Fiz. 102 (1995) 420] [hep-th/9409117] [INSPIRE].
X. Bekaert, M. Rausch de Traubenberg and M. Valenzuela, An infinite supermultiplet of massive higher-spin fields, JHEP 05 (2009) 118 [arXiv:0904.2533] [INSPIRE].
T. Basile, X. Bekaert and N. Boulanger, Mixed-symmetry fields in de Sitter space: a group theoretical glance, JHEP 05 (2017) 081 [arXiv:1612.08166] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
M.S. Costa and T. Hansen, AdS weight shifting operators, JHEP 09 (2018) 040 [arXiv:1805.01492] [INSPIRE].
M.S. Drew and J.D. Gegenberg, Conformally covariant massless spin-2 field equations, Nuovo Cim. A 60 (1980) 41 [INSPIRE].
A.O. Barut and B.-W. Xu, On conformally covariant spin-2 and spin-3/2 equations, J. Phys. A 15 (1982) L207.
S. Deser and R.I. Nepomechie, Anomalous propagation of gauge fields in conformally flat spaces, Phys. Lett. B 132 (1983) 321 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
W. Siegel, All free conformal representations in all dimensions, Int. J. Mod. Phys. A 4 (1989) 2015 [INSPIRE].
R.R. Metsaev, All conformal invariant representations of d-dimensional anti-de Sitter group, Mod. Phys. Lett. A 10 (1995) 1719 [INSPIRE].
A. Iorio, L. O’Raifeartaigh, I. Sachs and C. Wiesendanger, Weyl gauging and conformal invariance, Nucl. Phys. B 495 (1997) 433 [hep-th/9607110] [INSPIRE].
J. Erdmenger and H. Osborn, Conformally covariant differential operators: symmetric tensor fields, Class. Quant. Grav. 15 (1998) 273 [gr-qc/9708040] [INSPIRE].
A.Y. Segal, Conformal higher spin theory, Nucl. Phys. B 664 (2003) 59 [hep-th/0207212] [INSPIRE].
V.K. Dobrev, Invariant differential operators and characters of the AdS4 algebra, J. Phys. A 39 (2006) 5995 [hep-th/0512354] [INSPIRE].
R. Marnelius, Lagrangian higher spin field theories from the O(N) extended supersymmetric particle, arXiv:0906.2084 [INSPIRE].
R.R. Metsaev, Gauge invariant two-point vertices of shadow fields, AdS/CFT, and conformal fields, Phys. Rev. D 81 (2010) 106002 [arXiv:0907.4678] [INSPIRE].
M.A. Vasiliev, Bosonic conformal higher-spin fields of any symmetry, Nucl. Phys. B 829 (2010) 176 [arXiv:0909.5226] [INSPIRE].
X. Bekaert and M. Grigoriev, Higher order singletons, partially massless fields and their boundary values in the ambient approach, Nucl. Phys. B 876 (2013) 667 [arXiv:1305.0162] [INSPIRE].
R.R. Metsaev, Long, partial-short, and special conformal fields, JHEP 05 (2016) 096 [arXiv:1604.02091] [INSPIRE].
S.M. Kuzenko and M. Ponds, Conformal geometry and (super)conformal higher-spin gauge theories, JHEP 05 (2019) 113 [arXiv:1902.08010] [INSPIRE].
M. Flato and C. Fronsdal, One massless particle equals two Dirac singletons: elementary particles in a curved space. 6, Lett. Math. Phys. 2 (1978) 421 [INSPIRE].
M. Flato and C. Fronsdal, On Dis and Racs, Phys. Lett. B 97 (1980) 236 [INSPIRE].
C. Fronsdal, Flat space singletons, Phys. Rev. D 35 (1987) 1262 [INSPIRE].
E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals Math. 40 (1939) 149 [Nucl. Phys. B Proc. Suppl. 6 (1989) 9] [INSPIRE].
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Ponomarev, D. 3d conformal fields with manifest sl(2, ℂ). J. High Energ. Phys. 2021, 55 (2021). https://doi.org/10.1007/JHEP06(2021)055
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DOI: https://doi.org/10.1007/JHEP06(2021)055